How does one interpret the distribution over parameters in bayesian estimation? I am new to Bayesian estimation. The assumption that the parameters are random variables seems a little unsettling to me. For example when considering a model for data, what physical interpretation can I provide to the equation
$$
\begin{eqnarray*}
P(Data) & = & \sum_{\theta} {P(Data,\theta)} \\
        & = & \sum_{\theta} {P(Data|\theta)*P(\theta)}
\end{eqnarray*}
$$
This $P(\theta)$ i.e. probability over parameters, seems to be a bit awkward, after all how do I know what is the relative probability of the process of generation being a gaussian MM with this particular parameter combination instead of say a neural network with that parameter configuration. 
And further it is intuitive to think of one process generating the data, whose parameters we are guessing. But instead here we have multiple processes generating the data in tandem, i.e. a sense of a true model is lost.
 A: 
how do I know what is the relative probability of the process of generation being a gaussian MM with this particular parameter combination instead of say a neural network with that parameter configuration.

Your $\theta$ is the set of parameters in your model.  So for a Gaussian mixture model they are the means, covariances, and mixing parameters.  In a Neural Network they are the weights and biases.  These are totally different sets of quantities, so there's no reason to think that the $P(\theta)$ in either case will be related, either a priori or after seeing $D$.  
$P(D \mid \theta)$ is the part of the formula that will be realised as a mixture model or a network, or whatever.  But you have to decide, otherwise your prior is for the wrong quantities, which makes no sense.

And further it is intuitive to think of one process generating the data, whose parameters we are guessing. But instead here we have multiple processes generating the data in tandem, i.e. a sense of a true model is lost.

You already think of the data as being potentially generated by different values of $\theta$ before any Bayesian questions arise.  After all, the likelihood tells you how likely the data would have been generated under different sets of values.  But your 'in tandem' idea suggests you think they all do it 'all at once' in the Bayesian case, so there is no sense of 'one true model'.  That's a mistake.  Maybe think of it like this:
Call the 'true model parameters' $\theta_0$.  Bayesians and everybody else can agree that these are the things we want to know about.  Then $D$ is actually a sample from $P(D \mid \theta_0)$.  We just don't happen to know the $\theta_0$ is.
Our $P(D \mid \theta)$, where $\theta$ is any setting of parameters, just specifies the mechanism by which $D$ is assumed to be generated if we knew what the parameters were - a 'forward model' if you like.  Often it's straightforwardly physical, think of the $\theta$ as settings in a control panel.  Bayesian methods start with $P(\theta)$ - your opinions or knowledge about what $\theta_0$ might be before seeing $D$, and then condition on $D$ to get $P(\theta \mid D)$ - your new opinions or knowledge about what $\theta_0$ is after seeing $D$.  
The sum you present above is actually mostly useful just as a normalising constant on the way to getting $P(\theta \mid D)$ which actually is useful.  It's our updated beliefs about $\theta_0$.  It has some other roles, as 'evidence', but for the purposes of your question these aren't relevant.
A: This was too long for the comments, so posting it here.
From what the others have pointed out about thinking about the prior as a belief, I think a road-block in understanding had been combining the prior and the conditional.
The prior $P(\theta)$ is understood as a belief in what the true $\theta$ might be. The conditional $P(Data|\theta)$ is better thought in frequentist terms, i.e. take a model with this $\theta$ and generate many samples from it, and just count the frequencies for each sample. Their combination $\sum_{\theta} {P(\theta)\times P(Data|\theta)}$ doesn't remain a concrete process with a well defined $\theta$. So the problem is to understand that.
Suppose, initially I didn't have any concrete data, I just had a belief about what the background generating process could be, i.e. a $P(\theta)$. Also, for each process I could tell what the frequencies $P(Data|\theta)$ would be. Because I wasn't really sure about the process, the $P(Data)$ was a belief: With all my uncertainty about $\theta$, I'd on average expect data, if I ever collected any, to have a distribution like this $P(Data)$. 
But now I actually collect some samples, call this set $S$, and I calculate the frequencies of the samples. What I have now is $P(Data|S)$. 
But I could write: 
$P(Data|S)=\sum_{\theta} P(Data|\theta)P(\theta|S)$. Thinking in this way, my counting probability $P(Data|S)$ has been arrived by first changing my belief about $\theta$ to $P(\theta|S)$, which becomes more spiked towards a particular $\theta$, and the data distribution now looks more like $P(Data|\theta)$ for that $\theta$. So, was the crux the difference between $P(Data)$ and $P(Data|S)$?
